We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.
We show that discrete exponentials form a basis of discrete holomorphic functions on a finite critical map. On a combinatorially convex set, the discrete polynomials form a basis as well.
For a locally symmetric space , we define a compactification which we call the “geodesic compactification”. It is constructed by adding limit points in to certain geodesics in . The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give for locally symmetric spaces. Moreover, has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...
On a graph, we give a characterization of a parabolic Harnack inequality and Gaussian estimates for reversible Markov chains by geometric properties (volume regularity and Poincaré inequality).